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This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. $\textit{Digital halftoning}$ is a family of printing technologies for getting full color images from only a few different colors deposited at dots all of the same size. The simplest version consist in obtaining grey scale images from only black and white dots. A corollary of the theorem is that for $\textit{error diffusion}$, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.
This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+\kappa$. We prove that the boundary (in $\mathbb{C}$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon, and that $\infty$ is not accessible through any nonhyperbolic ("queer") stable component. The main part of the argument consists of demonstrating a general "Squeezing Lemma", which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.
We give examples of infinitely renormalizable quadratic polynomials $F_c: z\mapsto z^2+c$ with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrary close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one. The argument is based upon a new tool, a "Recursive Quadratic Estimate" for the Poincaré series of an infinitely renormalizable map.
We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the hairiness phenomenon", there exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent $\delta_\mathrm{cr}$ is equal to the hyperbolic dimension $\operatorname{HD}_\mathrm{hyp}(J(f))$. Moreover, if $\operatorname{area} J(f)=0$ then $\operatorname{HD}_\mathrm{hyp} (J(f))=\operatorname{HD}(J(f))$. In the stationary case, the last statement can be reversed: if $\operatorname{area} J(f)> 0$ then $\operatorname{HD}_\mathrm{hyp} (J(f))< 2$. We also give a new construction of conformal measures on $J(f)$ that implies that they exist for any $\delta\in [\delta_\mathrm{cr}, \infty)$, and analyze their scaling and dissipativity/conservativity properties.
In the paper we discuss two questions about smooth expanding dynamical systems on the circle. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequence of asymptotic length ratios are precisely those given by a positive Hölder continuous function $s$ (solenoid function) on the Cantor set $C$ of $2$-adic integers satisfying a functional equation called the matching condition. The functional equation for the $2$-adic integer Cantor set is $$ s (2x+1)= \frac{s (x)} {s (2x)} \left( 1+\frac{1}{ s (2x-1)}\right)-1. $$ We also present a one-to-one correspondence between solenoid functions and affine classes of $2$-adic quasiperiodic tilings of the real line that are fixed points of the 2-amalgamation operator. (ii) We calculate the precise maximum possible level of smoothness for a representative of the system, up to diffeomorphic conjugacy, in terms of the functions $s$ and $cr(x)=(1+s(x))/(1+(s(x+1))^{-1})$. For example, in the Lipschitz structure on $C$ determined by $s$, the maximum smoothness is $C^{1+\alpha}$ for $0 < \alpha \le 1$ if, and only if, $s$ is $\alpha$-H\"older continuous. The maximum smoothness is $C^{2+\alpha}$ for $0 < \alpha \le 1$ if, and only if, $cr$ is $(1+\alpha)$-H\"older. A curious connection with Mostow type rigidity is provided by the fact that $s$ must be constant if it is $\alpha$-H\"older for $\alpha > 1$.
The object of this paper is to prove some general results about rational idempotents for a finite group $G$ and deduce from them geometric information about the components that appear in the decomposition of the Jacobian variety of a curve with $G-$action.
We give an algorithm to find explicit primitive rational idempotents for any $G$, as well as for rational projectors invariant under any given subgroup. These explicit constructions allow geometric descriptions of the factors appearing in the decomposition of a Jacobian with group action: from them we deduce the decomposition of any Prym or Jacobian variety of an intermediate cover, in the case of a Jacobian with $G-$action. In particular, we give a necessary and sufficient condition for a Prym variety of an intermediate cover to be such a factor.
Let $(M,J,\Omega)$ be a polarized complex manifold of Kähler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of Kähler metrics that are invariant under $G$ and represent the cohomology class $\Omega$, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the $L^2$-norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its only fixed points, or extremal solitons, are extremal metrics. We prove local time existence of the flow, and conclude that if the lifespan of the solution is finite, then the supremum of the norm of its curvature tensor must blow-up as time approaches it. We end up with some conjectures concerning the plausible existence and convergence of global solutions under suitable geometric conditions.
Global results are proved about the way in which Boyland's forcing partial order organizes a set of braid types: those of periodic orbits of Smale's horseshoe map for which the associated train track is a star. This is a special case of a conjecture introduced in [1], which claims that forcing organizes all horseshoe braid types into linearly ordered families which are, in turn, parametrized by homoclinic orbits to the fixed point of code 0.
This paper investigates the set of angles of the parameter rays which land on the real slice [-2, 1/4] of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for the tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as biaccessible points of quadratic Julia sets are given.
In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of $\mathbb{C}^2$. The general problem is also reduced to the study of the combinatorics of certain resolutions of singularities. Whereas we deal with $\mathbb{C}$-complete vector fields, our results also apply to $\mathbb{R}$-complete ones thanks to a theorem of Forstneric [Fo].